direct product, abelian, monomial, 2-elementary
Aliases: C2×C108, SmallGroup(216,9)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C108 |
| C1 — C2×C108 |
| C1 — C2×C108 |
Generators and relations for C2×C108
G = < a,b | a2=b108=1, ab=ba >
Smallest permutation representation of C2×C108
►Regular action on 216 points
Generators in S216
(1 162)(2 163)(3 164)(4 165)(5 166)(6 167)(7 168)(8 169)(9 170)(10 171)(11 172)(12 173)(13 174)(14 175)(15 176)(16 177)(17 178)(18 179)(19 180)(20 181)(21 182)(22 183)(23 184)(24 185)(25 186)(26 187)(27 188)(28 189)(29 190)(30 191)(31 192)(32 193)(33 194)(34 195)(35 196)(36 197)(37 198)(38 199)(39 200)(40 201)(41 202)(42 203)(43 204)(44 205)(45 206)(46 207)(47 208)(48 209)(49 210)(50 211)(51 212)(52 213)(53 214)(54 215)(55 216)(56 109)(57 110)(58 111)(59 112)(60 113)(61 114)(62 115)(63 116)(64 117)(65 118)(66 119)(67 120)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 134)(82 135)(83 136)(84 137)(85 138)(86 139)(87 140)(88 141)(89 142)(90 143)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 151)(99 152)(100 153)(101 154)(102 155)(103 156)(104 157)(105 158)(106 159)(107 160)(108 161)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108)(109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216)
G:=sub<Sym(216)| (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,169)(9,170)(10,171)(11,172)(12,173)(13,174)(14,175)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,183)(23,184)(24,185)(25,186)(26,187)(27,188)(28,189)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)>;
G:=Group( (1,162)(2,163)(3,164)(4,165)(5,166)(6,167)(7,168)(8,169)(9,170)(10,171)(11,172)(12,173)(13,174)(14,175)(15,176)(16,177)(17,178)(18,179)(19,180)(20,181)(21,182)(22,183)(23,184)(24,185)(25,186)(26,187)(27,188)(28,189)(29,190)(30,191)(31,192)(32,193)(33,194)(34,195)(35,196)(36,197)(37,198)(38,199)(39,200)(40,201)(41,202)(42,203)(43,204)(44,205)(45,206)(46,207)(47,208)(48,209)(49,210)(50,211)(51,212)(52,213)(53,214)(54,215)(55,216)(56,109)(57,110)(58,111)(59,112)(60,113)(61,114)(62,115)(63,116)(64,117)(65,118)(66,119)(67,120)(68,121)(69,122)(70,123)(71,124)(72,125)(73,126)(74,127)(75,128)(76,129)(77,130)(78,131)(79,132)(80,133)(81,134)(82,135)(83,136)(84,137)(85,138)(86,139)(87,140)(88,141)(89,142)(90,143)(91,144)(92,145)(93,146)(94,147)(95,148)(96,149)(97,150)(98,151)(99,152)(100,153)(101,154)(102,155)(103,156)(104,157)(105,158)(106,159)(107,160)(108,161), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216) );
G=PermutationGroup([[(1,162),(2,163),(3,164),(4,165),(5,166),(6,167),(7,168),(8,169),(9,170),(10,171),(11,172),(12,173),(13,174),(14,175),(15,176),(16,177),(17,178),(18,179),(19,180),(20,181),(21,182),(22,183),(23,184),(24,185),(25,186),(26,187),(27,188),(28,189),(29,190),(30,191),(31,192),(32,193),(33,194),(34,195),(35,196),(36,197),(37,198),(38,199),(39,200),(40,201),(41,202),(42,203),(43,204),(44,205),(45,206),(46,207),(47,208),(48,209),(49,210),(50,211),(51,212),(52,213),(53,214),(54,215),(55,216),(56,109),(57,110),(58,111),(59,112),(60,113),(61,114),(62,115),(63,116),(64,117),(65,118),(66,119),(67,120),(68,121),(69,122),(70,123),(71,124),(72,125),(73,126),(74,127),(75,128),(76,129),(77,130),(78,131),(79,132),(80,133),(81,134),(82,135),(83,136),(84,137),(85,138),(86,139),(87,140),(88,141),(89,142),(90,143),(91,144),(92,145),(93,146),(94,147),(95,148),(96,149),(97,150),(98,151),(99,152),(100,153),(101,154),(102,155),(103,156),(104,157),(105,158),(106,159),(107,160),(108,161)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108),(109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216)]])
C2×C108 is a maximal subgroup of
C4.Dic27 Dic27⋊C4 C4⋊Dic27 D54⋊C4 D108⋊5C2
216 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 9A | ··· | 9F | 12A | ··· | 12H | 18A | ··· | 18R | 27A | ··· | 27R | 36A | ··· | 36X | 54A | ··· | 54BB | 108A | ··· | 108BT |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 108 | ··· | 108 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
216 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C9 | C12 | C18 | C18 | C27 | C36 | C54 | C54 | C108 |
| kernel | C2×C108 | C108 | C2×C54 | C2×C36 | C54 | C36 | C2×C18 | C2×C12 | C18 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 6 | 8 | 12 | 6 | 18 | 24 | 36 | 18 | 72 |
Matrix representation of C2×C108 ►in GL3(𝔽109) generated by
| 1 | 0 | 0 |
| 0 | 108 | 0 |
| 0 | 0 | 108 |
| 100 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 77 |
G:=sub<GL(3,GF(109))| [1,0,0,0,108,0,0,0,108],[100,0,0,0,38,0,0,0,77] >;
C2×C108 in GAP, Magma, Sage, TeX
C_2\times C_{108} % in TeX
G:=Group("C2xC108"); // GroupNames label
G:=SmallGroup(216,9);
// by ID
G=gap.SmallGroup(216,9);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-3,-3,72,122,118]);
// Polycyclic
G:=Group<a,b|a^2=b^108=1,a*b=b*a>;
// generators/relations
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